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15   THE LEBESGUE-NIKODYM THEOREM           191

This shows first of all that iff is /z-negligible then v(/) = 0. Passing to the
quotient by the subspace Ji of /^-negligible functions, the linear form v on
jjf R(X) therefore defines a linear form /»-» v(/) on the subspace «#* R(X) of
LR(X, u) which is the canonical image of ^TR(X) in L«(X, /*). The inequality
(13.5.1.1) shows that this linear form is continuous (5.5.1) on the subspace
jf R(X); since the latter is dense in Lg(X, u) (13.11.6), the linear form/W v(/)
extends by continuity to a linear form on the Hilbert space LR(X, ^) (5.5.4).
Hence there exists a function g e J2?R(X, /j) such that v(/) = X#/) for all
/e jfH(X) (6.3.2). Since # is locally //-integrable (13.1 3), it follows that v =g • fi.

(II)   General case.   There exists a partition of X into a sequence of com-
pact sets Kn and a /^-negligible set N (13.9.2). If we put M = X - N = (J Kn ,

n

then (pM is locally /j-integrable and 1 — <pM is /^-negligible, hence (13.14.4)
<pM • in = /z. Put /zn = <pKn • /z and vn = <pKn • v. Since <pKn is the lower envelope
of a decreasing sequence of functions ^0 belonging to «^*R(X), the relation
v ^ fi implies vn ^ p,n for all n; furthermore, vn is bounded (13.14.4). Hence,
by (I) above, there exists a locally /vintegrable function gn such that
*»=&,-/*„ = (0n<PKn) ' M = \ffn<PKn\ ' V ((13.13.5) and (13.14.5)), and the func-
tion gn <pKn is locally /i-integrable. Let g denote the function which is equal
to \gn9&n\ °n each Kn and is zero on N. Then g is the sum of the series

00

Z \0n(PKn\- If /is any function ^0 in JfR(X), and m is any integer, then

f/w                \               m   r                               m   r

\f    ZI0-PKJ UA*=S       /I^^KJ^=Z
J      \n=l           /          n=l J                       n=l J

Kndv£ f

/dv,
and therefore ((13.8.4) and (13.13.1)) the function # is locally //-integrable.

00

Also, since /= £/%„ almost everywhere (with respect to ju), we have

n- 1

(13.8.4) \fg d/j, = I fcpM dv. To show that v = g • /j, it remains to be proved
that N is also ^-negligible. But by definition (1 3.5.5), for each e > 0 there exists
a function h e ,/ such that (p^^h and X^) ^ fi- Since // is the upper envelope
of an increasing sequence of functions ^0 belonging to J>TR(X), the inequality
v <* /z implies that v*(/z) ^ /z(/z) ^ e, and the proof is complete.

The following lemma will be generalized later (13.15.8):

(13.15.2)   If(^j)i^j<,r is any finite sequence of complex measures on X, there
exists a positive measure 1 such that each /Zy is a measure with base Ly series (Problem 3) and x belongs to